# Multi-Objective Parametric Shape Optimisation of Body-Centred Cubic Lattice Structures for Additive Manufacturing

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Design Methodology

^{®}for the finite element analysis of the lattice structures and the optimisation of the geometry through single-/multi-objective genetic algorithm optimisations. Figure 3 presents a flowchart of the proposed lattice design approach.

#### 2.1. Implicit Surface Model of BCC Structure

^{®}to create iso-surfaces representing the corresponding lattice geometry. In order to achieve symmetric features on both sides of the strut members within a lattice, the construction process began with the creation of a single strut member. This was accomplished by mirroring the geometry of a strut, which was defined by Equation (2), around the outer end of the strut. A symmetrical model is important to avoid sharp corners on both sides of the strut, i.e., the same filet radius defined for the struts merging at the centre of the cell was defined for the opposite sides of the struts located at the corners of the unit cell. The unit cell was subsequently tessellated in a 3D space, with a spatial distribution defined by the parameters nx, ny, and nz representing the number of unit cells along the $x-axis$, $y-axis$, and $z-axis$, respectively. In this work, $nx=ny=nz=4$ was set for creating test samples for the numerical analysis and experimental validation, as demonstrated in Figure 5. The generated iso-surface model of the BCC lattice structure was then converted into a STL model for further finite element analysis (FEA), optimisation, and 3D printing. The process included extracting surface points corresponding to the iso-surface model. Subsequently, these surface points were triangulated to produce a triangular mesh, which could be saved as an STL file.

#### 2.2. Finite Element Analysis

^{®}using the MATLAB PDE toolbox. The boundary conditions corresponding to the FE model are presented in Figure 6. The bottom surface of the lattice structure model was fixed in the vertical (z-axis) direction, while it was allowed to move freely (frictionless) in the x-y horizontal plane. The top surface of the lattice was subject to a displacement loading of 2 mm downward (-z-axis), producing an overall strain of ${\epsilon}_{o}=5\%$. A static structural FEA was performed to study the linear elastic behaviour of lattice structures under uni-axial compression. The total reaction force on the top surface of the lattice geometry was derived from FEA. The reaction force was divided by the face area (40 × 40 mm) of the lattice sample to calculate the overall stress generated, which was then divided by the applied strain $\left({\epsilon}_{o}=5\%\right)$ to find the elastic modulus ${E}_{l}$ of the lattice model. The relative elastic modulus ${E}_{r}$ of the lattice was calculated by dividing the elastic modulus of the lattice by the Young’s modulus of the building material (${E}_{s}$), i.e., ${E}_{r}=\raisebox{1ex}{${E}_{l}$}\!\left/ \!\raisebox{-1ex}{${E}_{s}$}\right.$. The von-Mises stress (${\sigma}_{vm}$) was also calculated across the lattice structure for the further investigation of the performance of different designs.

#### 2.3. Preliminary Investigation of the Effect of Design Variables on the Mechanical Response of the Lattice

_{r}, and f

_{r}, due to the scattered distribution of data points within the 3D plot. From Figure 11b, it can be seen that, in general, as ${\rho}_{r}$ increases, so does ${\sigma}_{vm}^{max}$ (it should be noted that the results are obtained under displacement loading, where the applied displacement remains constant for all lattice samples; however, the reaction force varies, resulting in different stress levels within the lattice structures). However, compared to the stiffness–density data presented earlier in Figure 10, the stress–density data appears more scattered over a wider range. This indicates the importance of carefully adjusting the design variables to achieve the desired response from the lattice structure. In any case, the preliminary investigation of the effect of design variables on lattice structure properties demonstrates the significance and necessity of the implementation of an appropriate optimisation strategy for the design of the lattice structure, particularly for multi-objective optimisation problems, for instance, when the lattice structure is designed for maximum stiffness and minimum von-Mises stress with a target volume fraction (relative density) constraint. This was further investigated in the following section.

#### 2.4. Parametric Shape Optimisation of BCC Structures through the Multi-Objective Genetic Algorithm

^{®}to optimise two or more objectives simultaneously. MOGA begins by a random creation of an initial population of candidate solutions (individuals or chromosomes), which are then evaluated in terms of their performance (fitness) based on the objective functions of the problem. A non-dominated sorting technique is then used to rank the chromosomes based on their dominance relationship. Selection operators, such as tournament or roulette wheel selections, are used by MOGA to choose the best fit chromosomes. It also uses genetic operators, including crossover and mutation, to create offspring for the next generation. The offspring, along with some individuals from the population at present, replaces the previous generation, and the process is repeated until a termination criterion, such as a maximum number of generations or achieving a desired level of convergence, is met. Throughout the evolutionary process, MOGA maintains a set of non-dominated solutions, the solutions that cannot be improved in any objective without sacrificing the performance of another, known as the Pareto front. This set represents the trade-off between conflicting objectives offering decision makers a diverse set of solutions to choose from, based on their preferences.

#### 2.5. Single-Objective Genetic Algorithm Optimisation (SOGA)

#### 2.6. Numerical Comparison of the Solutions

## 3. Additive Manufacturing and Mechanical Testing

^{3}.

^{®}computer program, which captured readings at a frequency of 20 readings per second. To visually document the deformation induced by the compression plate of the machine, a high-speed, high-resolution camera capable of capturing 60 frames per second and performing high-definition video recording was utilised. An illustrative depiction of a BCC structure undergoing compression testing is presented in Figure 15.

^{2}) under compression, resulting in the computation of global stress values (σ). Plotting these stress values (σ) against the strain (ε) induced by the compression test machine facilitated the determination of the Young’s modulus of the lattice material (${E}_{l}$), based on the slope of the linear elastic region within the stress–strain curve. Similarly, following ASTM D695-15 [41], the Young’s modulus (${E}_{s}$) of solid samples sized 10.0 mm × 10.0 mm × 10.0 mm was calculated. These solid samples were 3D-printed using the same material and settings as those used for the lattice samples. This process yielded the stress–strain curves depicted in Figure 16, from which a Young’s modulus of ${E}_{s}=104.3$ MPa was computed for the bulk material. Parallel to the numerical calculations presented in Section 2, the experimental relative modulus values (${E}_{r}$) for each lattice structure were deduced, employing the formula ${E}_{r}=\raisebox{1ex}{${E}_{l}$}\!\left/ \!\raisebox{-1ex}{${E}_{s}$}\right.$.

## 4. Results and Discussion

#### 4.1. Comparison of Different Designs

#### 4.2. Insights into the Failure Modes of DLP-Printed BCC Lattices

#### 4.3. Further Remarks on the Generalisation of the Lattice Structure Design for AM

## 5. Summary and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Classical model of a BCC lattice unit cell with cylindrical members of uniform thickness at a 45° angle; (

**b**) example of a shape-optimised BCC unit cell.

**Figure 4.**Demonstration of the approach for merging implicit surfaces by multiplying equations of individual cylinders.

**Figure 5.**The procedure used for the construction of the proposed implicit surface-based BCC lattice unit cell and tessellation into a $4\times 4\times 4$ lattice structure.

**Figure 6.**Boundary conditions applied to a lattice structure and the resultant reaction forces on the top surface of the lattice.

**Figure 9.**(

**a**) The relation between t

_{r}, f

_{r}, and ρ

_{r}. (

**b**) The relationship between t

_{r}, f

_{r}, and E

_{r}.

**Figure 10.**Demonstration of stiffness–density (${E}_{r}$-${\rho}_{r}$) relationship obtained from 900 different designs and the corresponding best fit equations (

**a**) covering the full relative density range, and (

**b**) focusing on the 0–30% relative density range.

**Figure 11.**Data points corresponding to 900 different designs of the BCC lattice structure presenting (

**a**) the relationship between the maximum von-Mises stress and design variables (t

_{r}, f

_{r}), and (

**b**) the maximum von-Mises stress–relative density (${\sigma}_{vm}^{max}$-${\rho}_{r}$) relationship.

**Figure 12.**(

**a**) MOGA Pareto front presenting a set of optimised solutions with respect to objective 1 (${E}_{r}$) and objective 2 (${\sigma}_{vm}^{max}$); (

**b**) distribution of objectives corresponding to the 900 different designs studied in Section 2.3.

**Figure 13.**(

**a**) Stiffness–density (${E}_{r}$-${\rho}_{r}$) graph for the set of solutions presented by the MOGA Pareto front; (

**b**) Poisson’s ratio ($\upsilon $) against relative density ${\rho}_{r}$ for the set of solutions presented by the MOGA Pareto front.

**Figure 14.**Evolution histories corresponding to two runs of SOGA: (

**a**) maximising objective 1 ${E}_{r}$ (minimising $\frac{1}{{E}_{r}}$) exclusively, and (

**b**) minimising objective 2 ${\sigma}_{vm}^{max}$, exclusively.

**Figure 15.**(

**a**) Demonstration of a BCC lattice specimen during compression test and (

**b**) different stages of the compression test presented from left to right.

**Figure 16.**Stress–strain plots corresponding to five solid samples fabricated through the DLP process resulting in an average Young’s modulus of 104.3 MPa.

**Figure 17.**Stress–strain plots associated with the compression testing of a set of samples containing BCC

_{C}, BCC

_{M}, BCC

_{E}, and BCC

_{vm}specimens.

**Figure 18.**Comparison of the stiffness of different BCC structures. Relative elastic modulus (${E}_{r}$) calculated numerically (${E}_{FEA}$) and experimentally (${E}_{exp}$) used as an indicator of stiffness. The experimental results are based on the average measurements obtained from three fabricated samples for each distinct design configuration.

**Figure 19.**Failure modes of classical and optimised BCC structures: (

**a**) classical BCC at lattice structure level, (

**b**) classical BCC at unit cell level, (

**c**) classical BCC at strut level, (

**d**) optimised BCC at lattice structure level, (

**e**) optimised BCC at unit cell level, (

**f**) optimised BCC at strut level.

**Table 1.**Different design variants of the proposed implicit-based BCC lattice structure achieved by increasing ${f}_{r}$ while keeping ${t}_{r}$ constant at 0.1.

${f}_{r}=4.0$ | ${f}_{r}=8.0$ | ${f}_{r}=12.0$ | ${f}_{r}=16.0$ | ${f}_{r}=20.0$ | ${f}_{r}=24.0$ |

**Table 2.**Different design variants of the proposed implicit-based BCC lattice structure achieved by increasing ${t}_{r}$ while assigning lower band values to ${f}_{r}$ from ${f}_{r}=12\ast {t}_{r}$.

${t}_{r}=0.2\phantom{\rule{0ex}{0ex}}{f}_{r}=2.4$ | ${t}_{r}=0.4\phantom{\rule{0ex}{0ex}}{f}_{r}=4.8$ | ${t}_{r}=0.6\phantom{\rule{0ex}{0ex}}{f}_{r}=7.2$ | ${t}_{r}=0.8\phantom{\rule{0ex}{0ex}}{f}_{r}=9.6$ | ${t}_{r}=1.0\phantom{\rule{0ex}{0ex}}{f}_{r}=12$ | ${t}_{r}=1.2\phantom{\rule{0ex}{0ex}}{f}_{r}=14.4$ |

**Table 3.**Performance comparison of different BCC structures optimised through MOGA and SOGA as well as the classical model of BCC.

$\mathbf{Classical}\mathbf{BCC}\mathbf{\left(}BC{C}_{C}\mathbf{\right)}$ | $\mathrm{MOGA}-\mathrm{Optimised}\mathrm{BCC}\mathrm{for}\mathrm{Both}{E}_{r}$ $\mathrm{and}{\sigma}_{vm}^{max}$$(BC{C}_{M}$) | $\mathrm{SOGA}-\mathrm{Optimised}\mathrm{BCC}\mathrm{for}{E}_{r}$$(BC{C}_{E}$) | $\mathrm{SOGA}-\mathrm{Optimised}\mathrm{BCC}\mathrm{for}{\sigma}_{vm}^{max}$$(BC{C}_{vm}$) | |
---|---|---|---|---|

Design variables | $-$ | ${t}_{r}=0.178\phantom{\rule{0ex}{0ex}}{f}_{r}=9.338$ | ${t}_{r}0.179\phantom{\rule{0ex}{0ex}}{f}_{r}=9.338$ | ${t}_{r}=0.100\phantom{\rule{0ex}{0ex}}{f}_{r}=10.500$ |

${\sigma}_{vm}$ contour | ||||

${\sigma}_{vm}^{max}$ (MPa) | 8.8892 | 6.7772 | 6.8542 | 6.6617 |

${E}_{r}$ | 0.00811 | 0.01098 | 0.01113 | 0.01071 |

$\mathbf{Classical}\mathbf{BCC}\mathbf{\left(}BC{C}_{C}\mathbf{\right)}$ | $\mathrm{MOGA}-\mathrm{Optimised}\mathrm{BCC}\mathrm{for}\mathrm{Both}{E}_{r}$ $\mathrm{and}{\sigma}_{vm}^{max}$$(BC{C}_{M}$) | $\mathrm{SOGA}-\mathrm{Optimised}\mathrm{BCC}\mathrm{for}{E}_{r}$$(BC{C}_{E}$) | $\mathrm{SOGA}-\mathrm{Optimised}\mathrm{BCC}\mathrm{for}{\sigma}_{vm}^{max}$$(BC{C}_{vm}$) | |
---|---|---|---|---|

Unit cell | ||||

DLP-printed lattice sample | ||||

Mass (g) | 16.95 | 16.47 | 17.45 | 15.60 |

${\rho}_{r}$ | 0.22 | 0.22 | 0.23 | 0.21 |

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**MDPI and ACS Style**

Ali, H.M.A.; Abdi, M.
Multi-Objective Parametric Shape Optimisation of Body-Centred Cubic Lattice Structures for Additive Manufacturing. *J. Manuf. Mater. Process.* **2023**, *7*, 156.
https://doi.org/10.3390/jmmp7050156

**AMA Style**

Ali HMA, Abdi M.
Multi-Objective Parametric Shape Optimisation of Body-Centred Cubic Lattice Structures for Additive Manufacturing. *Journal of Manufacturing and Materials Processing*. 2023; 7(5):156.
https://doi.org/10.3390/jmmp7050156

**Chicago/Turabian Style**

Ali, Hafiz Muhammad Asad, and Meisam Abdi.
2023. "Multi-Objective Parametric Shape Optimisation of Body-Centred Cubic Lattice Structures for Additive Manufacturing" *Journal of Manufacturing and Materials Processing* 7, no. 5: 156.
https://doi.org/10.3390/jmmp7050156